Regularity of Powers of Edge Ideals: from Local Properties to Global Bounds

REGULARITY OF POWERS OF EDGE IDEALS: FROM LOCAL PROPERTIES TO GLOBAL BOUNDS

ARINDAM BANERJEE, SELVI KARA BEYARSLAN, AND HUY TAI` HA`

Abstract. Let I = I(G) be the edge ideal of a graph G. We give various general upper bounds for the regularity function reg Is, for s ≥ 1, addressing a conjecture made by the authors and Alilooee. When G is a gap-free graph and locally of regularity 2, we show that reg Is = 2s for all s ≥ 2. This is a weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function reg Is, for s ≥ 1, via local information of I.

1. Introduction

During the last few decades, studying the regularity of powers of homogeneous ideals has evolved to be a central research topic in algebraic geometry and commutative algebra. This research program began with a celebrated theorem, proved independently by Cutkosky-Herzog-Trung [9] and Kodiyalam [24], which stated that for a homogeneous ideal I in a standard graded algebra over a ﬁeld, the regularity function reg Is is asymptotically a linear function (see also [3, 33]). Though despite much eﬀort from many researchers, this asymptotic linear function is far from being well understood. In this paper, we investigate this regularity function for edge ideals of graphs. We shall explore several classes of graphs for which this regularity function can be explicitly described or bounded in terms of combinatorial data of the graphs. This problem has been studied recently by many authors (cf. [1, 2, 4, 5, 6, 12, 13, 21, 22, 23, 27, 30]). Our initial motivation for this paper is the general belief that global conclusions often could be derived from local information. Particularly, local conditions on an edge ideal I (i.e., conditions on reg(I : x), for x ∈ V (G)) should give a global understanding of the function reg Is, for s ≥ 1. Our motivation furthermore comes from the following conjectures (see [5, 28, 29]), which provide a general upper bound for the regularity function of edge ideals, and describe a special class of edge ideals whose powers (at least 2) all have linear arXiv:1805.01434v3 [math.AC] 14 Apr 2019 resolutions.

Conjecture A (Alilooee-Banerjee-Beyarslan-H`a).Let G be a simple graph with edge ideal I = I(G). For any s ≥ 1, we have reg Is ≤ 2s + reg I − 2.

Conjecture B (Nevo-Peeva). Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and reg I = 3. Then, for all s ≥ 2, we have reg Is = 2s. 1 Our aim is to investigate Conjectures A and B using the local-global principle. Finding general upper bounds for reg I(G)s has received a special interest and generated a large number of papers during the last few years. This partly thanks to a general lower bound for reg I(G)s given in [6]; particularly, if ν(G) denotes the induced matching number of G then, for any s ≥ 1, we have reg I(G)s ≥ 2s + ν(G) − 1. (1.1) Our ﬁrst main result gives a weaker general upper bound for reg I(G)s than that of Conjecture A. The motivation of this result comes from an upper bound for the regularity of I(G) given by Adam Van Tuyl and the last author, namely reg I(G) ≤ β(G) + 1, where β(G) denotes the matching number of G (see [16]). We prove the following theorem. Theorem 3.4. Let G be a graph with edge ideal I = I(G), and let β(G) be its matching number. Then, for all s ≥ 1, we have reg Is ≤ 2s + β(G) − 1.

As a consequence of Theorem 3.4, for the class of Cameron-Walker graphs, where ν(G) = β(G), we have reg Is = 2s + ν(G) − 1 ∀ s ≥ 1.

A graph G is said to be locally of regularity at most r − 1 if reg(I(G): x) ≤ r − 1 for all vertex x in G. Note that, by [8, Proposition 4.9], if G is locally of regularity at most r − 1 then reg I(G) ≤ r. In the local-global spirit, we reformulate Conjecture A to a slightly weaker conjecture as follows.

Conjecture A0. Let G be a simple graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have reg Is ≤ 2s + r − 2.

Our next main result proves Conjecture A0 for gap-free graphs. Theorem 4.2. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have reg Is ≤ 2s + r − 2.

It is an easy observation that if I(G)s has a linear resolution for some s ≥ 1 then G must be gap-free. Conjecture B serves as a converse statement to this observation, and has remained intractable. By applying the local-global principle, we prove a weaker statement, in which the condition reg I = 3 is replaced by the condition that G is locally linear (i.e., locally of regularity at most 2). Our main result toward Conjecture B is stated as follows. Theorem 4.5. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally linear. Then for all s ≥ 2, we have reg Is = 2s. 2 As a consequence of Theorem 4.5, we quickly recover a result of Banerjee, which showed that if G is gap-free and cricket-free then I(G)s has a linear resolution for all s ≥ 2 (see Corollary 4.6). We end the paper by exhibiting an evidence for Conjecture A0 at the first nontrivial value of s, i.e., s = 2, for all graphs. Theorem 5.1. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then, for any edge e ∈ E(G), reg(I2 : e) ≤ r. Particularly, this implies that reg(I2) ≤ r + 2. Our paper is structured as follows. In the next section we give necessary notation and terminology. The reader who is familiar with previous work in this research area may want to proceed directly to Section 3. In Section 3, we discuss general upper bound for the regularity function, aiming toward Conjecture A. Theorem 3.4 is proved in this section. In Section 4, we focus further on gap-free graphs, investigating both Conjectures A0 and B using the local-global principle. Theorems 4.2 and 4.5 are proved in this section. We end the paper with Section 5, proving Theorem 5.1 and discussing briefly how an effective bound on the regularity of I(G)2 may give us information on the regularity of the second symbolic power I(G)(2). This gives a glimpse into future work on the regularity function of symbolic powers of edge ideals. Acknowledgement. Part of this work was done while the first named and the last named authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). We would like to express our gratitude toward VIASM for its support and hospitality. The last named author is partially supported by Simons Foundation (grant #279786) and Louisiana Board of Regents (grant #LEQSF(2017-19)-ENH-TR-25). The authors thank Thanh Vu for pointing out a mistake in our first version of the paper.

2. Preliminaries

In this section, we collect notations and terminology used in the paper. For unexplained notions, we refer the reader to standard texts [7, 11, 18, 26, 31, 34]. Graph Theory. Throughout the paper, G shall denote a ﬁnite simple graph with vertex set V (G) and edge set E(G). A subgraph G0 of G is called induced if for any two vertices 0 0 u, v in G , uv ∈ E(G ) ⇔ uv ∈ E(G). For a subset W ⊆ V (G), we shall denote by GW the induced subgraph of G over the vertices in W , and denote by G − W the induced subgraph of G on V (G) \ W . When W = {w} consists of a single vertex, we also write G − w for G − {w}. The complement of a graph G, denoted by Gc, is the graph on the same vertex set V (G) in which uv ∈ E(Gc) ⇔ uv 6∈ E(G). Deﬁnition 2.1. Let G be a graph.

(1)A walk in G is a sequence of (not necessarily distinct) vertices x1, x2, . . . , xn such that xixi+1 is an edge for all i = 1, 2, . . . , n. A circuit is a closed walk (i.e., when x1 ≡ xn). (2)A path in G is a walk whose vertices are distinct (except possibly the ﬁrst and the last vertices). 3 (3)A cycle in G is a closed path. A cycle consisting of n distinct vertices is called an n-cycle and often denoted by Cn. (4) An anticycle is the complement of a cycle.

A graph in which there is no induced cycle of length greater than 3 is called a chordal graph. A graph whose complement is chordal is called a co-chordal graph. Deﬁnition 2.2. Let G be a graph. (1)A matching in G is a collection of disjoint edges. The matching number of G, denoted by β(G) is the maximum size of a matching in G. (2) An induced matching in G is a matching C such that the induced subgraph of G over the vertices in C does not contain any edge other than those already in C. The induced matching number of G, denoted by ν(G), is the maximum size of an induced matching in G. Deﬁnition 2.3. Let G be a graph. (1) Two disjoint edges uv and xy are said to form a gap in G if G does not have an edge with one endpoint in {u, v} and the other in {x, y}. (2) If G has no gaps then G is called gap-free. Equivalently, G is gap-free if and only if c ν(G) = 1 (i.e., G contains no induced C4).

For any integer n, Kn denotes the complete graph over n vertices (i.e., there is an edge connecting any pair of vertices). For any pair of integers m and n, Km,n denotes the complete bipartite graph; that is, a graph with a bipartition (U, V ) of the vertices such that |U| = m, |V | = n and E(Km,n) = {uv | u ∈ U, v ∈ V }. Deﬁnition 2.4.

(1) A graph isomorphic to K1,3 is called a claw. A graph without any induced claw is called a claw-free graph. (2) A graph isomorphic to the graph with vertex set {w1, w2, w3, w4, w5} and edge set {w1w3, w2w3, w3w4, w3w5, w4w5} is called a cricket. A graph without any induced cricket is called a cricket-free graph. Observation 2.5. A claw-free graph is cricket-free. Notation 2.6. Let G be a graph, let u, v ∈ V (G), and let e = xy ∈ E(G).

(1) The set of vertices incident to u, the neighborhood of u, is denoted by NG(u). Set NG[u] = NG(u) ∪ {u}. (2) The set of vertices incident to an endpoint of e, the neighborhood of e, is denoted by N (e). Set N [e] = N (e) ∪ {x, y}. G G G (3) The degree of u is degG(u) = NG(u) . An edge is called a leaf or a whisker if any of its vertices has degree exactly 1. (4) The distance between u and v, denoted by d(u, v), is the fewest number of edges that must be traversed to travel from u to v in G.

We can naturally extend these notions to get NG(W ), NG[W ], NG(E) and NG[E] for a subset of the vertices W ⊆ V (G) or a subset of the edges E ⊆ E(G). 4 Deﬁnition 2.7. Let G be a graph. (1) A collection W of the vertices in G is called an independent set if there is no edge connecting two vertices in W . (2) The independent complex of G, denoted by ∆(G), is the simplicial complex whose faces are independent sets of G.

Commutative Algebra. Let G be a simple graph over the vertices V (G) = {x1, . . . , xn}. By abusing notation, we shall identify the vertices of G with the variables in a polynomial ring S = k[x1, . . . , xn], where k is any inﬁnite ﬁeld. Particularly, we shall use uv to denote both the edge uv in G and the monomial uv in S (the choice would be obvious from the context).

Deﬁnition 2.8. Let G be a graph over the vertices V (G) = {x1, . . . , xn}. The edge ideal of G is deﬁned to be I(G) = hxy | xy ∈ E(G)i ⊆ S.

Castelnuovo-Mumford regularity is the invariant being investigated in this paper. We shall give a definition most suitable for our context. Definition 2.9. Let S be a standard graded polynomial ring over a field k. The regularity of a finitely generated graded S module M, written as reg M, is given by

reg(M) := max{j − i| Tori(M, k)j 6= 0}.

For a graph G, we shall use reg I(G) and reg G interchangeably. The following simple bound is often used without references. Lemma 2.10 (See [15, Lemma 3.1]). Let G be a simple graph and let H be an induced subgraph of G. Then reg I(H) ≤ reg I(G). Particularly, for any vertex v ∈ V (G), we have that reg I(G − v) ≤ reg I(G).

A standard use of short exact sequences yields the following result, which we shall also often use. Lemma 2.11. Let I ⊆ S be a monomial ideal, and let m be a monomial of degree d. Then reg I ≤ max{reg(I : m) + d, reg(I, m)}. Moreover, if m is a variable appearing in I, then reg I is equal to one of the right-hand-side terms. Deﬁnition 2.12. Let r ∈ N. A graph G is said to be locally of regularity ≤ r if for every vertex x ∈ V (G), we have reg(I(G): x) ≤ r. A graph which is locally of regularity ≤ 2 is called locally linear.

Auxiliary Results. We next recall a few results that are useful for our purpose. We shall make use of the following characterization for edge ideals of graphs with linear resolutions. This characterization was ﬁrst given in topological language by Wegner [35] and later, independently, by Lyubeznik [25] and Fr¨oberg [14] in monomial ideals language. 5 Theorem 2.13 (See [14, Theorem 1]). Let G be a simple graph. Then reg I(G) = 2 if and only if G is a co-chordal graph.

In the study of powers of edge ideals, Banerjee developed the notion of even-connection and gave an important inductive inequality in [4]. This inductive method has proved to be quite powerful, which we shall make use of often. Theorem 2.14. For any ﬁnite simple graph G and any s ≥ 1, let the set of minimal s monomial generators of I(G) be {m1, ...., mk}, then s+1 s+1 s reg I(G) ≤ max{reg(I(G) : ml) + 2s, 1 ≤ l ≤ k, reg I(G) }.

The ideal (I(G)s+1 : m) in Theorem 2.14 and its generators are understood via the following notion of even-connection. Deﬁnition 2.15. Let G = (V,E) be a graph. Two vertices u and v (u may be the same as v) are said to be even-connected with respect to an s-fold product e1 ··· es where ei’s are edges of G, not necessarily distinct, if there is a path p0p1 ··· p2k+1, k ≥ 1 in G such that:

(1) p0 = u, p2k+1 = v. (2) For all 0 ≤ l ≤ k − 1, p p = e for some i. 2l+1 2l+2 i (3) For all i, {l ≥ 0 | p2l+1p2l+2 = ei} ≤ {j | ej = ei} . (4) For all 0 ≤ r ≤ 2k, prpr+1 is an edge in G.

It turns out that (I(G)s+1 : m) is generated by monomials in degree 2. Theorem 2.16 ([4, Theorem 6.1 and Theorem 6.7]). Let G be a graph with edge ideal I = I(G), and let s ≥ 1 be an integer. Let m be a minimal generator of Is. Then (Is+1 : m) is minimally generated by monomials of degree 2, and uv (u and v may be the same) is a minimal generator of (Is+1 : m) if and only if either {u, v} ∈ E(G) or u and v are even- connected with respect to m.

3. General Upper Bounds for Regularity Function

The aim of this section is to give a weaker general upper bound for reg I(G)s than that of Conjecture A. The heart of many studies on regularity of powers of edge ideals is to understand the s colon ideal J = I(G) : e1 . . . es−1 in making use of Banerjee’s inductive method, Theorem 2.14. We start by examining a local property for J.

Lemma 3.1. Let G be a simple graph with edge ideal I = I(G) and let s ∈ N. Let s 0 e1, . . . , es−1 ∈ E(G), J = I : e1 . . . es−1, and let G be the graph associated to the polarization of J. Let w ∈ V (G).

s−1 (1) If e1 is a leaf of G then J = I : e2 . . . es−1. (2) Suppose that w 6∈ NG[{e1, . . . , es−1}]. Then s J : w = I(G − NG[w]) : e1 . . . es−1 + (u u ∈ NG[w]). 6 (3) Suppose that w ∈ NG[e1]. Then t J : w = (I(G − NG0 [w]) : f1 . . . ft−1) + (u u ∈ NG0 (w))

for some t ≤ s, and a subcollection {f1, . . . , ft−1} of {e2, . . . , es−1}. Moreover, in t this case, the graph associated to the polarization of I(G − NG0 [w]) : f1 . . . ft−1 is an t induced subgraph of that associated to the polarization of I(G − NG[w]) : f1 . . . ft−1.

Proof. (1) It follows from Theorem 2.16 that J is obtained by adding to I quadratic generators uv, where u and v are even-connected in G with respect to e1 . . . es−1. If e1 is an isolated edge then clearly, by deﬁnition, the even-connected path between u and v does not s−1 contain e1. Thus, uv ∈ I : e2 . . . es−1 and (1) is proved.

(2) It can be seen that if w 6∈ NG[{e1, . . . , es−1}] then w is not in any even-connected path with respect to e1 . . . es−1. Thus, even-connected paths with respect to e1 . . . es−1 between two vertices that are not in NG[w] are even-connected path with respect to e1 . . . es−1 in G − N [w]. Furthermore, any edge uv ∈ J, for which u ∈ N [w] (similarly if v ∈ N [w]), G G G would be divisible by u ∈ J : w and, thus, subsumed into the ideal (u u ∈ NG[w]). Therefore, (2) follows.

u even-connected v even-connected w

ei1 e1 eip

Figure 1. When w ∈ e1

u even-connected w v even-connected

ei1 e1 eip

Figure 2. When w ∈ NG(e1)

(3) We ﬁrst observe that for any subcollection {f1, . . . , ft−1} of {e1, . . . , es−1} (for some t ≤ e), by the deﬁnition of even-connection, we have t I(G − NG0 [w]) : f1 . . . ft−1 ⊆ J ⊆ (J : w).

Moreover, for any u ∈ NG0 (w), u and w are even-connected with respect to e1 . . . es−1, and so uw ∈ J, i.e., u ∈ (J : w). Thus, we have the inclusion t (I(G − NG0 [w]) : f1 . . . ft−1) + (u u ∈ NG0 (w)) ⊆ (J : w). 7 To prove the other inclusion, let us analyse the minimal generators of (J : w) more closely. Consider any uv ∈ J, where u and v are even-connected with respect to e1 . . . es−1. If v ≡ w (similarly if u ≡ w) then u ∈ N 0 (w). If u, v 6≡ w, but v ∈ N 0 (w) (similarly if u ∈ N 0 (w)), G G G then uv is subsumed in the ideal (u u ∈ NG0 (w)).

Suppose now that u, v 6∈ NG0 [w]. Then u, v ∈ G − NG0 [w], which are even-connected with respect to e1 . . . es−1. Observe that if the even-connected path between u and v contains e1 then, by considering a subpath of this path, either u and w or v and w are even-connected with respect to e . . . e (see Figures 1 and 2). That is, either u or v is in N 0 (w), and so 1 s−1 G uv is again subsumed in the ideal (u u ∈ NG0 (w)). Therefore, we may assume that u and v are even-connected with respect to a subcollection {f1, . . . , ft−1} of {e2, . . . , es−1}. That is, t uv ∈ I(G − NG0 [w]) : f1 . . . ft−1. w even-connected u even-connected v

even-connected

0 Figure 3. When an even-connected path u — v contains w ∈ NG0 [w]

To establish the last statement, consider any two vertices u and v which are even-connected in G − NG[w] with respect to f1 . . . ft−1. If the even-connected path between u and v does not contain any vertex in NG0 [w] \ NG[w] then u and v are even-connected in G − NG0 [w]. If 0 the even-connected path between u and v contain a vertex w ∈ NG0 [w] \ NG[w] (see Figure 3) then, by combining with the even-connected path from w to w0, either u and w or v and w 0 are even-connected in G . That is, either u or v is already in NG0 [w] (or equivalently, not in t G−NG0 [w]). Hence, the graph associated to the polarization of I(G−NG0 [w]) : f1 . . . ft−1 is t an induced subgraph of that associated to the polarization of I(G − NG[w]) : f1 . . . ft−1.

By understanding local properties of J in Lemma 3.1, we are able to give a general upper bound for the regularity function based on a well chosen numerical function on families of graphs. Speciﬁc interesting general bounds can be obtained by picking these numerical functions suitably. Deﬁnition 3.2. A collection F of simple graphs is a hierarchy if for any nonempty graph G ∈ F, both G − u and G − NG[u] are in F for any vertex u ∈ V (G). Theorem 3.3. Let F be a hierarchy family of simple graphs. Let f : F −→ N be a function satisfying the following properties: (1) for any G ∈ F, reg I(G) ≤ f(G); and (2) for any nonempty graph G ∈ F and each non-isolated vertex w ∈ V (G),

f(G − w) ≤ f(G) and f(G − NG[w]) ≤ max{f(G) − 1, 2}. 8 Then, for any G ∈ F and any s ≥ 1, we have reg I(G)s ≤ 2s + f(G) − 2.

Proof. Fix a graph G ∈ F and let I = I(G). If f(G) ≤ 2 then the result is immediate from [19]. Assume that f(G) ≥ 3. Then the condition on f(G − NG[w]) reads f(G − NG[w]) ≤ f(G) − 1. By Theorem 2.14 and the hypothesis that reg I(G) ≤ f(G), it suﬃces to show that for any collection of edges e1, . . . , es−1 in G (not necessarily distinct), we have s reg(I : e1 . . . es−1) ≤ f(G). (3.1) s We shall prove (3.1) by induction on s and on the size of the graph G. Let J = I : e1 . . . es−1. The statement is trivial if s = 1 (whence, J = I) or if G is the empty graph (whence, J = (0)). Suppose that s ≥ 2 and G is not the empty graph. Let w ∈ V (G) be any vertex in G. It follows from Lemma 3.1 that reg(J : w) is equal to s s either reg(I(G − NG[w]) : e1 . . . es−1) or reg(I(G − NG0 [w]) : e1 . . . es−1) where the graph t associated to the polarization of I(G − NG0 [w]) : f1 . . . ft−1 is an induced subgraph of that t associated to the polarization of I(G − NG[w]) : f1 . . . ft−1. If the latter is the case, then by Lemma 2.10 and the fact that polarization does not change the regularity, we have t reg(J : w) ≤ reg(I(G − NG[w]) : f1 . . . ft−1).

Thus, since G − NG[w] ∈ F, by induction on the size of the graphs and our assumption, we have

reg(J : w) ≤ f(G − NG[w]) ≤ f(G) − 1 for any vertex w ∈ V (G). (3.2)

By taking, for example, a vertex cover of the graph associated to the polarization of J, we may assume that we have a collection of distinct vertices w1, . . . , wl of G such that (J, w1, . . . , wl) = (w1, . . . , wl). Observe that for each i = 1, . . . , l − 1, we have

(J, w1, . . . , wi): wi+1 = (J : wi+1) + (w1, . . . , wi). Thus, by [17, Corollary 3.2] and (3.2), we get

reg[(J, w1, . . . , wi): wi+1] ≤ reg(J : wi+1) ≤ f(G) − 1.

This, by successively applying Lemma 2.11 with (J, w1, . . . , wi) and wi+1, implies that

reg(J, w1) ≤ f(G).

The assertion now follows by utilizing Lemma 2.11 with J and w1.

Based on the known upper bound for reg I(G), given in [16], one can take f(G) in Theorem 3.3 to be the matching number of a graph and obtain the following interesting bound for the regularity function. Theorem 3.4. Let G be a simple graph with edge ideal I = I(G). Let β(G) denote the matching number of G. Then, for all s ≥ 1, we have reg Is ≤ 2s + β(G) − 1. 9 Proof. Let F be the family of all simple graphs. Then F clearly is a hierarchy. Let f(G) = β(G) + 1 for all G ∈ F. It is easy to see that: (1) reg I(G) ≤ f(G) by [16]; and (2) For any non-isolated vertex w in G, clearly β(G − w) ≤ β(G), and we can always add an edge incident to w to any matching of G − NG[w] to get a bigger matching, and so f(G − NG[w]) ≤ f(G) − 1.

Hence, the statement follows from Theorem 3.3.

A particular interesting application of Theorem 3.4 is for the class of Cameron-Walker graphs introduced in [10]. These are graphs for which ν(G) = β(G). See [20] for a further classiﬁcation of Cameron-Walker graphs. Corollary 3.5. Let G be a Cameron-Walker graph and let I = I(G) be its edge ideal. Then, for all s ≥ 1, we have reg Is = 2s + ν(G) − 1.

Proof. The conclusion is an immediate consequence of Theorem 3.4 noting that ν(G) = β(G) if G is a Cameron-Walker graph.

It is known, by the main theorem of [19], that if I(G) has a linear resolution then so does I(G)s for any s ∈ N. Thus, the ﬁrst nontrivial case of Conjecture A is for those graphs G such that G is locally linear and reg I(G) > 2. Recall that by [8, Proposition 4.9], in this case, we necessarily have reg I(G) = 3. Theorem 3.3 allows us to settle Conjecture A for this class of graphs. Theorem 3.6. Let G be a graph with edge ideal I = I(G). Suppose that G is locally linear. Then for all s ≥ 1, we have reg Is ≤ 2s + reg I − 2 ≤ 2s + 1.

Proof. Let F be the family of locally linear graphs (including those whose edge ideals have linear resolutions). Define f : F −→ N by f(G) = reg I(G) for all G ∈ F. By the definition and Lemma 2.10, the edge ideal of any proper induced subgraph of G ∈ F has a linear resolution. Thus, F is a hierarchy and f satisfies conditions of Theorem 3.3. The conclusion now follows from that of Theorem 3.3. Example 3.7. Let G be a graph such that Gc is triangle-free (see, for example, Figure 4). It can be seen that for any x ∈ V (G), G−NG[x] is a complete graph (and, thus, is of regularity 2). Therefore, G is a locally linear graph.

4. Regularity Function of Gap-free Graphs

In this section, we focus on gap-free graphs, investigating both Conjectures A0 and B. We start with a stronger version of [4, Lemma 6.18]. The proof is almost the same as that given in [4, Lemma 6.18] 10 x1 x2 x3

x4 x5 Figure 4. A graph whose complement is triangle-free

Lemma 4.1. Let G be a gap-free graph with edge ideal I = I(G). Let e1, . . . , es−1 be a col- s 0 lection of edges, let J = I : e1 . . . es−1, and let G be the graph associated to the polarization of J. Let W ⊆ V (G). Suppose that u = p0, . . . , p2k+1 = v is an even-connected path in G with respect to e1 . . . es−1 satisfying: (1) u, v 6∈ W ; and (2) this path is of the longest possible length with respect to condition (1).

0 Then G − W − NG0 [u] is obtained by adding isolated vertices to an induced subgraph of G − NG[u].

Proof. By Theorem 2.16, uv ∈ G0 −W . Consider any other edge u0v0 ∈ G0 \G with u0, v0 6∈ W . 0 0 Then, there is an even-connected path u = q0, . . . , q2l+1 = v in G with respect to e1 . . . es−1 for some 1 ≤ l ≤ k.

If there exist i and j such that p2i+1p2i+2 and q2j+1q2j+2 are the same edge in G then by combining these two even-connected paths, either u0 or v0 will be even-connected to u. That 0 0 0 is, either u or v will become an isolated vertex in G − W − NG0 [u]. We may assume that the two even-connected path between u, v and u0, v0 do not share any edge.

Consider p1p2 and q1q2. Since these two edges do not form a gap in G, they must be connected. Let us now explore diﬀerent possibilities for this connection. 0 0 If p1 ≡ q1 then u and v are even-connected with respect to e1 . . . es−1, and so v becomes 0 an isolated vertex in G − W − NG0 [u]. If p1 ≡ q2 (similarly for the case that p2 ≡ q1) then 0 0 u and u are even-connected with respect to e1 . . . es−1, and so u becomes an isolated vertex 0 in G − W − NG0 [u]. 0 0 If p1q1 ∈ E(G) then combining the two even-connected paths between u, v and u , v and 0 the edge p1q1, we get an even-connected path between v and v that is of length > k, a contradiction. If p1q2 ∈ E(G) (similarly for the case that p2q1 ∈ E(G)) then by combining 0 0 the two even-connected paths between u, v and u , v and the edge p1q2, we have an even connected path between u0 and v that is of length > k, a contradiction. 0 0 0 Thus, in any case, either u or v will becomes an isolated vertex in G −W −NG0 [u]. That 0 0 is, any edge in G \ G will reduce to an isolated vertex in G − W − NG0 [u]. The statement is proved. Our next main result establishes Conjecture A0 for gap-free graphs. 11 Theorem 4.2. Let G be a graph with edge ideal I = I(G) and let r ≥ 3 be an integer. Assume that G is gap-free and locally of regularity ≤ r − 1. Then, for all s ∈ N, we have reg Is ≤ 2s + r − 2.

Proof. By [8, Proposition 4.9], we have reg I ≤ r. By Theorem 2.14, it suﬃces to show that for any collection of edges e1, . . . , es−1 (not necessarily distinct) in G, we have s reg(I : e1 . . . es−1) ≤ r.

0 s Let G be the graph associated to the polarization of J = I : e1 . . . es−1. It follows from Lemma 2.11 that, for any vertex x ∈ G0, 0 0 0 reg G ≤ max{reg(G − NG0 [x]) + 1, reg(G − x)}. (4.1) 0 0 Thus, we shall show that reg(G − x) ≤ r and reg(G − NG0 [x]) ≤ r − 1.

Let u and v be even-connected in G with respect to e1 . . . es−1 such that the even-connected 0 0 path u = p0, . . . , p2k1+1 = v is of maximum possible length. By Lemma 4.1, G − NG [u] is obtained by adding isolated vertices to an induced subgraph of G − NG[u]. Thus, by Lemma 0 2.10, we have reg(G − NG0 [u]) ≤ reg(G − NG[u]) ≤ r − 1. It remains to consider reg(G0 − u). Let u0 and v0 be even-connected in G with respect to 0 0 0 0 0 e1 . . . es−1 such that u , v ∈ G −u and there is an even-connected path u = q0, . . . , q2l+1 = v in G with respect to e1 . . . es−1 such that l is the maximum possible length. By using Lemma 0 0 0 4.1 again, we can deduce that reg(G − u − NG0 [u ]) ≤ reg(G − NG[u ]) ≤ r − 1. Thus, by applying (4.1) to the graph G0 − u, it suﬃces to show that reg(G0 − {u, u0}) ≤ r. We can continue in this fashion until all edges in G0 \ G are examined, i.e., we obtain a collection W ⊆ V (G) such that G0 − W = G − W , and reduce the problem to showing that reg(G0 − W ) = reg(G − W ) ≤ r. This is obviously true by Lemma 2.10 and the fact that reg G ≤ r. The theorem is proved.

We shall now shift our attention to Conjecture B. We begin by an improved statement of [8, Corollary 6.5]. Lemma 4.3. Let G be a gap-free and cricket-free graph. Then G is locally linear.

Proof. We may assume that G contains no isolated vertices. By Theorem 2.13, it suﬃces to c show that (G \ NG[x]) is chordal for any vertex x in G. Note that since G \ NG[x] is an induced subgraph of G, it is gap-free and cannot have any induced anticycle of length 4.

Suppose that W = {w1, w2, . . . , wn} is such that G[W ] is an anticycle of length n ≥ 5 in G \ NG[x]. Clearly, W ∩ NG[x] = ∅. Let y be a neighbor of x. Since G is gap-free, {x, y} and {w1, w3} cannot form a gap. Thus, these edges must be connected in G. That is, either {y, w1} or {y, w3} (or both) must be an edge in G.

Suppose that {y, w1} and {y, w3} are both edges in G. Then, by considering edges {x, y} and {w2, wn} in G, either {y, w2} or {y, wn} must be an edge in G. If {y, w2} is an edge, then the induced subgraph on {x, y, w1, w2, w3} is a cricket in G, a contradiction. Other- wise, {y, wn} ∈ E(G). Since {x, y} and {w2, wn−1} cannot form a gap in G, we must have 12 {y, wn−1} ∈ E(G). Thus, the induced subgraph on {x, y, w1, wn−1, wn} is a cricket in G, a contradiction.

If {y, w1} ∈ E(G) and {y, w3} 6∈ E(G) (similarly for the case {y, w1} 6∈ E(G) and {y, w3} ∈ E(G)), then {y, wn} must be an edge in G; otherwise, {x, y} and {w3, wn} form a gap in G. By considering {x, y} and {w2, wn−1}, either {y, w2} or {y, wn−1} must be an edge in G. If {y, w2} ∈ E(G), then the induced subgraph on {x, y, w1, w2, wn} is a cricket in G, a contradiction. Otherwise, {y, wn−1} ∈ E(G), and the induced subgraph on {x, y, w1, wn−1, wn} is a cricket in G, a contradiction. Example 4.4. There are examples for locally linear gap-free graphs for which the regularity could be either 2 or 3 (see Figure 5).

x1 x4 x1 x5

C4 x2 C5 x4

x2 x3

x3 Figure 5. Locally linear gap-free graphs with regularity 2 and 3 (respectively)

On the other hand, note that if G is not gap-free, then ν(G) ≥ 2 =⇒ reg I(G) ≥ 3. Thus, if, in addition, I(G) is locally linear, then we have reg I(G) = 3 by [8, Proposition 4.9]. Figure 6 depicts such a graph.

x1 x5

x2 x3 x4 Figure 6. A graph that is not gap-free but locally linear with regularity 3

We are now ready to state our main result toward Conjecture B. In this result, we establish the conclusion of Conjecture B replacing the condition that reg I(G) = 3 by the condition that G is locally linear. Theorem 4.5. If G is a graph with edge ideal I = I(G). Suppose that G is gap-free and locally linear. Then, for all s ≥ 2, we have reg Is = 2s.

Proof. Again, by Theorem 2.14, it suﬃces to show that for any collection of edges e1, . . . , es−1 (not necessarily distinct), we have s reg(I : e1 . . . es−1) ≤ 2. 13 0 s That is, the graph G associated to the ideal J = I : e1 . . . es−1 is a co-chordal graph. By [4, Lemma 6.14], G0 is also gap-free, and so G0 does not contain an anticycle of length 0 4. Suppose that W = {w1 . . . wn}, for n ≥ 5, is such that G [W ] is an induced anticycle of G0. It follows from [4, Lemma 6.15] that G[W ] is an induced anticycle of G.

Let e1 = ab. We shall consider diﬀerent possibilities for the relative position of a and b with respect W . 0 If a, b ∈ W , say a ≡ w1 and b ≡ wi (for i 6= 1), then since {w1, w2}, {w1, wn} 6∈ E(G ), b 6= w2, wn. Consider the edges {a, b} and {w2, wn}. These do not form a gap (and a is not connected to neither w2 nor wn), and so either {b, w2} ∈ E(G) or {b, wn} ∈ E(G). If {b, w2} ∈ E(G) then w2 and w3 are even-connected with respect to e1 = ab, which implies 0 that {w2, w3} ∈ E(G ), a contradiction. If {b, wn} ∈ E(G) then wn−1 and wn are even- 0 connected with respect to e1 = ab, which implies that wn−1wn ∈ E(G ), also a contradiction.

If a ∈ W , say a = w1, and b 6∈ W (similar to the case where a 6∈ W and b ∈ W ) then by considering the edges {a, b} and {w2, wn} again, the same arguments as above would lead to a contradiction. If a, b 6∈ W and either a or b is not connected to any vertices in W , then G0[W ] (being also an anticycle in G) is an anticycle in either G − NG[a] or G − NG[b], which is a contradiction to the local linearity of G. It remains to consider the case that a, b 6∈ W , and both a and b are connected to W . Assume that aw1 ∈ E(G). Consider the pair of edges {a, b} and {w2, wn}. If either {b, w2} ∈ E(G) or {b, wn} ∈ E(G) then, as before, we would have either {w2, w3} ∈ E(G) or {wn−1, wn} ∈ E(G), which is a contradiction. Thus, we must have either {a, w2} ∈ E(G) or {a, wn} ∈ E(G). Without loss of generality, we may assume that {a, w2} ∈ E(G). We continue by considering the pair of edges {a, b} and {w3, wn}. A similar argument shows that {a, w3} ∈ E(G). We can keep going in this fashion to get {a, wi} ∈ E(G) for all i = 1, . . . , n − 2. Now, it can be seen that b cannot be connected to any of the wi without creating an even-connection that gives {wi, wi+1} ∈ E(G), for some i, which is a contradiction. We have shown that such a collection of the vertices W cannot exists. That is, G0 is a co-chordal graph. The theorem is proved.

Theorem 4.5 immediately recovers the following result of Banerjee [4].

Corollary 4.6 ([4, Theorem 6.7]). Let G be a gap-free and cricket-free graph. Then, for any s ≥ 2, we have reg I(G)s = 2s.

Proof. The conclusion follows from Lemma 4.3 and Theorem 4.5.

Example 4.7. Let 2K2 denote a gap and let K6 denote the complete graph on 6 vertices. Let G = 2K2 + K6 be the join of these two graphs (the join of two graphs H and K is obtained by taking the disjoint union of H and K and connecting each vertex in H with every vertex in K). Then, it can be seen G is locally linear but not gap-free. Particularly, it 14 follows that reg I(G)s 6= 2s for all s ∈ N. This gives an example of a locally linear graph G for which reg I(G)s 6= 2s for all s ∈ N.

5. Regularity of Second Powers of Edge Ideals

We end the paper with a ﬂavor of Conjecture A0 when s = 2. We also take a look at the symbolic square of edge ideals. Theorem 5.1. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then, for any edge e ∈ E(G), reg(I2 : e) ≤ r. Particularly, this implies that reg(I2) ≤ r + 2.

Proof. The second statement follows from the ﬁrst statement and Theorem 2.14. To prove the ﬁrst statement, we shall use induction on |V (G)|. Let J = I2 : e and let G0 be the graph associated to J. If there are no even-connected vertices in G with respect to e, then I2 : e = I, and the conclusion follows from [8, Proposition 4.9]. If there are edges in G0 which are not initially in G, then these edges are of the form xy where x ∈ N(a), y ∈ N(b) or xx0 where x ∈ N(a) ∩ N(b) and x0 is a new whisker vertex. Suppose that there exists at least one new edge of the form xy for x 6= y. Observe that J : x = I : x + (u | u ∈ N(b)). Thus reg(J : x) ≤ reg(I : x) ≤ r − 1. Furthermore, (J, x) = I(G \ x)2 : e. Therefore, by induction on |V (G)|, we have reg(J, x) ≤ r. Hence, by Lemma 2.11, we have reg J ≤ r. Suppose that the only new edges are of the form xx0, where x0 is a new whisker vertex. Observe that, in this case, J : x = I : x + (u | u ∈ N(a) ∪ N(b)) + (u0 | u0 is a whisker in the new edges ) (J, x) = I(G \ x)2 : e Thus, we also have reg(J : x) ≤ reg(I : x) ≤ r − 1 and reg(J, x) ≤ r by induction. Hence, by Lemma 2.11 again, we have reg J ≤ r. This completes the proof. Symbolic powers in general are much harder to handle than ordinary powers. The symbolic square of an edge ideal appears to be more tractable. We recall and rephrase a result from [32]. Theorem 5.2 ([32, Corollary 3.12]). For any graph G, (2) 2 I(G) = I(G) + (xixjxk | {xi, xj, xk} forms a triangle in G).

The last result of our paper is stated as follows. Theorem 5.3. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then reg(I(2)) ≤ r + 2.

(2) Proof. We ﬁrst note that, by Theorem 5.2, I ⊆ I. Let E(G) = {e1, . . . , el} and, for 0 ≤ i ≤ l, deﬁne (2) (2) Ji = (I + e1 ··· + ei):(ei+1) and Ki = (I + e1 ··· + ei). 15 Observe that Kl = I, and for all i we have the following short exact sequence. R R R 0 −→ (−2) −→ −→ −→ 0 (5.1) Ji Ki Ki+1

(2) This, particularly, implies that reg(I ) ≤ max {reg(Ji) + 2, reg I}. It follows from 1≤i≤l−1 Theorem 5.2 that 2 Ji = I : ei+1 + (xixjxk : ei+1 | {xi, xj, xk} forms a triangle in G).

Note that if e is an edge in the triangle {xi, xj, xk}, then (xixjxk : e) is a variable. If e shares a vertex with the triangle, then the colon ideal is generated by an edge and (xixjxk : e) ∈ I. If e and {xi, xj, xk} have no common vertices, then (xixjxk : e) = xixjxk ∈ I. Then, by 2 2 Theorem 2.16 we have Ji = I : ei+1 + (variables) and hence, reg Ji ≤ reg(I : e). The conclusion now follows from Theorem 5.1 and the use of [8, Proposition 4.9].

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Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, West Bengal, India E-mail address: [email protected] URL: https://http://maths.rkmvu.ac.in/~arindamb/

University of South Alabama, Department of Mathematics and Statistics, 411 University Boulevard North, Mobile, AL 36688-0002, USA E-mail address: [email protected] URL: https://sites.google.com/southalabama.edu/selvikara/home

Tulane University, Department of Mathematics, 6823 St. Charles Ave., New Orleans, LA 70118, USA E-mail address: [email protected] URL: http://www.math.tulane.edu/∼tai/